There are several methods known in the art for determining the fill fraction of a liquid in a container. For example, shaking a container of liquid by hand is a commonly used method to approximate the amount of liquid therein. This method is an empirical method since the method relies or is based solely on experiments or experience or is based on practical experience without reference to scientific principles. More accurate methods exist for measuring the container fill fraction. For example, a container supported by load cells could be filled with accurately measured quantities of liquid and vibrated over a known range of frequencies. This could establish liquid sloshing frequencies versus fill fraction or height of a liquid in the container.
Several other systems and methods for measuring the quantity of liquid in a container have been developed. U.S. Pat. No. 4,788,648, issued to Ferretti et al., discloses a method and system for measuring liquid level in a tank which determines values of differential pressure within the tank and, in accordance with the values of the differential pressure, calculates the instantaneous level of substance within tank. U.S. Pat. No. 4,815,323, issued to Ellinger et al., discloses an aircraft fuel quantity gauging method which uses ultrasonic signals. U.S. Pat. No. 4,901,245, issued to Olson et al., discloses a nonintrusive acoustic liquid level sensor system and method for detecting the level of a liquid in a tank. Also, U.S. Pat. No. 4,908,776, issued to Crill et al., discloses an apparatus and method for spacecraft fuel measurement which utilizes an accelerometer mounted proximal to the tank to measure sloshing of fuel in response to forces acting on the spacecraft. The actual amount of fuel remaining in the tank is estimated by comparing the measured frequency of oscillation with empirical data linking the frequency of oscillation with the amount of remaining fuel. The accuracy of this empirical method, however, is uncertain due to the effects of the gravitation acceleration (i.e., 32 ft/S.sup.2) and the mechanical constraints supporting the spacecraft, both of which are absent in orbit.
Other systems and methods are found in other patents, such as U.S. Pat. Nos. 4,908,783, 4,928,525 and 4,977,528.
An advantage of a systematic or theoretical model of prediction versus the empirical models is the elimination of expensive tests, measurement or calibration prior to use. Also, the accuracy of an empirical method based on ground testing is in doubt as previously noted.
One type of systematic method for predicting small amplitude slosh frequencies versus fill fractions for a more or less incompressible liquid that is not spinning in a uniform gravity field is to solve Laplace's equation with appropriate boundary conditions at the tank wall and free surface. At the tank wall, the liquid velocity normal to the wall must be zero and, at the free surface, the pressure must be constant and equal to the pressure of gas in the tank. A solution to this well known boundary condition problem can be obtained analytically for certain tank shapes, e.g., cylindrical, or by numerical methods, e.g., finite element analysis, for general geometries.
An equivalent pendulum model of the liquid can be deduced from this theory and can be used to accurately predict the sloshing forces on the container, provided it is not rotating. Using any one of a number of mechanical theories for multirigid body systems, this mechanical pendulum analog may be attached to a rigid body model of the container to predict the response of the overall container-liquid system. Such a model, which has been previously applied to determine the stability of rocket flight control systems, could also be used in the determination of fuel remaining.
When the container is rotating, equivalent pendulum models cannot completely describe the liquid behavior; and the general coupled system response is not obvious. Theoretical models treating the spinning, coupled, spacecraft-liquid system have so far been limited to predicting passive spacecraft stability at a single frequency (i.e., spacecraft nutation frequency). In addition, these methods are limited to on-axis cylindrical or axisymmetric tanks, are used in approximate Fourier expansion methods, or are highly inaccurate compared with air bearing tests.